I ntern. J. Math. Math. Si.
Vol. 4. No. 3 (1981) 503-512
503
PERMUTATION MATRICES AND MATRIX EQUIVALENCE OVER A
FINITE FIELD
GARY L. MULLEN
Department of Mathematics
The Pennsylvania State University
Sharon, Pennsylvania 16146
(Received March 21, 1980 and in revised form August 12, 1980)
Let F
ABSTRACT.
GF(q) denote the finite field of order q and F
the ring of
mxn
Let P
m x n matrices over F.
over F so that
Pn
n
be the set of all permutation matrices of order n
is ismorphic to S
A is equivalent to B relative to
tions 3 and
4, if
Pn
If
n
Pn
is a subgroup of
if there exists
PEPn
and A
such that AP
then
BEF
mxn
B. In sec-
formulas are given for the number of equivalence classes
of a given order and for the total number of classes.
In sections 5 and 6 we
study two generalizations of the above definition.
KEY WORDS AND PHRASES.
Permutation mtrix, equivalence, atomorphism,
field.
AMS(MOS) SUBJECT CLASSIFICATION CODES:
i.
finite
Primary 15A33, Secondary 12C99, 15A24.
INTRODUCTION.
In a series of papers [1-4,6-8] L. Carlitz, S. Cavior, and the author studied
use\.of
various forms of equivalence of functions over a finite field through the
permutation groups acting on the field itself.
matrices A and B to be equivalent if
bij
In [9] the author defined two
(aij)
(A) where (A) was
field while in [i0] B was said to be equivalent to A if B
computed by substitution.
of the
for some permutation
In the present paper we study another form of matrix
equivalence over a finite field through the use of permutation matrices and the
P61ya-deBruijn theory of enumeration.
Let F
and let F
mxn
GF(q) denote the finite field of order q
p
b
p is prime and b > 1
denote the ring of m x n matrices over F so that
..IFmxnl
q
mn
Let
504
Pn
G.L. MULLEN
be the set of all n x n matrices over F consisting entirely of zeros and ones
In the lit-
with the property that there is exactly one i in each row and column.
erature, such matrices have been called permutation
show that
P
It is not hard to
matrices.
is a group under matrix multiplication which is isomorphic to S
the symmetric group on n letters and consequently has order n’.
morphlsm can be defined as follows.
then define
peS n
p(1)
by
ei(i
If
PePn
the iso-
If
n).
i
Then
T:Pn
S
/
n
p
defined by T(P)
is an isomorphism.
2.
GENERAL THEORY.
If
is a subgroup of
DEFINITION I.
exists P e
Pn
we may make
If A, B e F
such that AP
then B is equivalent to A relative to
B.
This is an equivalence relation on F
.
the class of A relative to
by
so we let (A,) denote the order of
and let () be the total number of classes induced
If A,B e F
then B is equivalent to A relative to
mxn
THEOREM 2.1.
if there
Pn
if and
only if the columns of B are a permutation of the columns of A.
PROOF.
Suppose AP
the i in column
A becomes column
J
B where A
occurs in row
i]
J
(aij)P
(aiij)
J
of A is colunm
ij
of B.
we have a 1 in row i. and zeros elsewhere.
Then
3
A is equivalent to B.
COROLLARY 2.2.
det(B)
Then AP
In P suppose that for
J
l,...,n
so that column
J
of
of AP.
Conversely, suppose cqlumn
column
ij.
(aij).
If
,
Define P so that in
PeP
and AP
B so that
BeF
and B is equivalent to A relative to
nxn
then
+ det(A).
In fact, if AP
then det(B)
B and P corresponds to
det(A) while if
Sp
peS n
where
Sp
is an odd permutation then
is an even permutation
det(B)
-det(A).
PERMUTATION MATRICES AND MATRIX EQUIVALENCE
If A
DEFINITION 2.
P
Q and AP
then P is an automorp.hlsm
F
505
of A relative to
A.
If Aut(A,Q) denotes the set of all automorphlsms of A relative to
is easy to check that
where
[]
If A
Pn
such that AP
F
mxn
then for any subgroup
of P
n
.
(2.1)
let N(P,m,n,q) denote the number of m x n matrices A over GF(q)
A.
THEOREM 2.4.
If P corresponds to
pgS n
and
p
has (P) distinct cycles then
qm(P).
N(P,m,n,q)
PROOF.
Suppose the distinct cycles of
of A are identical.
p
are o
Using Theorem
O(p).
I
A if and only if within a given cycle of
2.1 it is clear that AP
p
the columns
The theorem then follows from the fact that a given column
can be constructed in q
3.
It is easy to prove
denotes the order of
If P E
then it
Aut(A,Q) is a group under matrix multiplication whose order
will be denoted by (A,Q).
THEOREM 2.3.
,
if
m
ways.
CYCLIC GROUPS.
lfil
<ps/t>.
<P> is a cyclic group of permutation matrices where
If
denote the subgroup of fl of order t where
eS
sponds to
let
P
t(P)
so that H(t)
denote-the number of cycles of
n,q) denotes the number of m
THEOREM 3.1.
t[s
pS/t
n matrices A over GF(q) such that
s, let H(t)
If P corre-
and suppose M(t,m
Aut(A,fl)
H(t).
For each divisor t of s
(P)
mat
V(a)q
M(t,m,n,q)
(3.1)
where (a) is the Mobius function.
PROOF.
By Theorem 2.4
q
m t (P)
GF(q) such that Aut(A,fl) < H(t).
tainment is proper.
counts the number of m x n matrices A over
From this we subtract those for which the con-
This number is given by
M(t,m,n,q)
where the sum is over all
uls, t lu
q
mt (P)
and t
u.
7.M(u,m,n,q),
(3.2)
After applying Mobius inversion
G.L. MULLEN
506
to
(3.2) we obtain (3.1).
For each divisor t of s there are tM(t,m,n,q)/s classes of
COROLLARY 3.2.
s/t and
I Y. t M(t,m,n,q).
%()
P=
<P> then
II
3.
(3.3)
t,ls
2, m
As an illustration, suppose q
so that if
s
3, and
n
0
i
0
0
i
0
01
i
0
One can easily check that M(3,3,3,2)
8 and
504 so that there are 168 classes of order 3, 8 classes of order 1
M(1,3,3,2)
and thus from (3.3), %()
176.
THE CASE
4.
In this section we consider the group Pn of all permutation matrices of order
P is isomorphic to S
the symmetric
n
n
We will employ the P61ya theory of enumeration to determine
n so that, as noted in the introduction,
group on n letters.
If
et of r elements.
of length t.
l,...,r b t denotes the number of cycles of
t
is called the
cycle
index of K.
k
(klk2"2
PSn (Xl’’’’’Xn)
I
+ 2k 2 +
+
nk
n
IKI -I
,xr)
PK(Xl
k
Pn
Suppose the permutation group K acts on a
b
b
b
1
2
r
where for
K consider the monomial x
x
...x
I
2
r
the number of classes induced by
2
b
b
Z x ix 2
I 2
The polynomial
b
r
x
r
(4.1)
It is well known [5] that
...knnkn)-l_,
k
x
k
1
I x2
2
kn where
...Xn
the sum is over all
n.
In the Plya theory of enumeration, let the domain D be the set of n columns
and let the range R be
RDI
Fmxnl.
q
[5, p. 157]
so that
the
set of q
m
possible column vectors so that
If K is a permutation group acting on D then elya’s theorem
states that the number of distinct classes is given by
%(Pn
Psn(qTM,... ,qm)
PK(IRI,..., IRI)
It follows directly from Theorem 2. i that %(P
is also the number of distributions of n indistinguishable objects into q
cells, or
(n +
qm-i)
n
so that we have proven
m
n
labelled
PERMUTATION MATRICES AND MATRIX EQUIVALENCE
Theorem 4.1.
If
%(Pn
(n +
n
mxn
Pn then
is the number of classes induced by
%(p
Suppose AEF
507
qm I).
n
has t distinct columns so that we have a partition of n with
I +...+ mt where each distinct column occurs mi times.
t
2.1 for each such A we have u(A,Pn)
so that by (2.1) v(A,Pn)
t parts say n
m
By Theorem
n
=i=l m..’l
(ml,...,mt).
The number of such A is the same as the number of functions from D into R whose
m
whose domain is of size n and whose preimage partition has
range is of size q
ml+...+mt
type
We may rewrite this with distinct m’s say
n.
where
Jml+...+jms
where
h(Jm1
t.
Jmlml+....+Jmsms=n
(qm)th(Jml ’Jm
Then the number of such functions is
s
’Jms
is the number of partitions of n of type
]mlml+.. "+]msms
n
and is given by Cauchy’s formula
’Jms
h(JmI
and
(qm) t
qm(qm-1)... (qm-t+l)
values to the partition blocks.
COROLLARY 4.2.
JmI
n./((mI
(Jml),.... (ms")
Jms
(Jm)"
s
is the falling factorial which assigns image
Hence we have proven
The number of classes induced by
m
(qt (JmI
t
Pn of order
(m1
n
,m s
is
’Jms
As an illustration of the above theory suppose q
2 and m
n
3 so that we
are considering the 512 3 x 3 matrices over GF(2) under the action of the symmetric
group S
Thus from Corollary 4.2 when t
3.
(8i)(i1
8 classes of order I, when t
8
2)
(2)(1,1
there are
3
(3)(3)
A(P3
(130)
3 so that there are
1+2 so that there are
3 we have n
56 classes of order 6 so that
Theorem 4.1 we also see that
5.
2 we have n
56 classes of order 3 and when t
8
i we have n
A(P3
I + i + I so that
120.
Moreover, from
120 classes.
A GENERALIZATION
In this section we generalize Definition I by considering a notion of matrix
equivalence which is similar to the idea of weak equivalence of functions over a
finite field considered by Cavior and the author in
group of m
2
m permutation matrices over GF(q).
is a subgroup of
Pn
we may make
If
[3] and [8].
i
Let P
is a subgroup of
m
be the
Pm and
508
G.L. MULLEN
If A
DEFINITION 3.
i
PEPn
2
such that QAP
B.
permutes the columns of A while
QEPm
if there exist Q E
Thus
then B is equivalent to A relative to
BEF
and P E
i
acts as a permutation group
on the domain D.
I
Clearly if
sections 3 and 4.
on the range R and
i
and
2
permutes the rows of A so that
2
is a permutation group acting
{Id.} we obtain the previous cases considered In
In this more general setting we will make use of the extended
P61ya theory of enumeration.
THEOREM 5.1.
groups
2
of D and
(Polya-deBruljn) The number of classes induced by permutation
of R is
i
PI
P2
Consider the q
row i, we have q
q
i-i
times.
m
m-i+l
a permutation group
0
(z2+z4+...)
Zl=Z2--...--0
sets where in each set one element of
For example, if q
(12...m).
2
possible column vectors of R in an m x q
i
0
i
0
i
0
i
0
0
i
i
0
0
i
I
0
0
0
0
i
i
I
i
By
m
array so that in
GF(q) is repeated
3 we list the 8 column vectors as
2 and m
0
Suppose now that
tion
Zl+Z2+’’"
is the cyclic group of order m generated by the permuta-
permute the rows of the m x q
letting
’I on the column vectors of the range R.
(123) then the column vectors
(el, C3, C5 ,C7, C2, C4,C6 ,C8).
(CI,...,C 8)
A
0
i
i
i
O
0
i
i
0
0
i
i
0
0
0
i
0
and Q is the permutation matrix
array, we induce
For example, if
of (5.2) are permuted to
If
i
m
[C
4c7c 6
509
PERMUTATION MATRICES AND MATRIX EQUIVALENCE
QA
then
0
i
I
1
o
].
1
1
0
[cc6c4].
Qg
By the isomorphism defined in section i,
tlon matrix Q.
corresponds to the permuta-
3
m
Q
Thus by applying
S
to the rows of the m x q
permutation on the column vectors of the range R.
array, we induce a
This is turn induces a permu-
tatlon of the rows of A which is equivalent to just permuting the rows of A by
Hence we can permute the rows of any matrix by
using the permutation matrix Q.
simply permuting the rows of the m x q
array.
is the cyclic group of prime order m acting on the q
i
If
m
column vectors
induced by a cyclic group of prime order m acting on the rows of the m x q
m
array,
it is not difficult to prove that
(xq
(Xl’’’’’xqm)
Pfl
m
+
(m-1)xq]xm (qm-q)/m)"
(5.3)
We are now ready to prove
THEOREM 5.2.
then if m
I is cyclic of prime order m and 2
If
n
i
mn
%(2 i
while if m.
y.
(t)
+ (m-l)q n/t
(qmn/ t
iron tlY’
(t)(qmn/t+(m-l)qn/t)
+
n
t#km
PROOF.
mn
If t
n/t eqm(zl+z2+’’"
zn/t
t
i (5.6) reduces to I/mn[q
In.
(t)/mn(q
If m
In
and I < t
M as before while if i < t
which
i
n
r
(t)q
mn/t
(s.5)
tl n
t=km
We must evaluate (5.1) which becomes for fixed
of n we have M
all t
(5.4)
ln
%(2’i
(t)
is cyclic of order n
+(m-l)e
q(zl+z2+...) e (qm-q)(Zm+Z2m+...)
(5.6)
zi=O.
mn
+ (m-l)qn].
mn/t + (m-l)q
# km for
n/t)
If m
n and t > i is a divisor
which proves (5.4) upon summing over
some positive integer k the (5.6) contributes
km for some k, (5.6) contributes ((t)q
mn/t )/n from
(5.5) follows.
As an illustration, suppose q
m
n
2 so that we are considering the 16
G.L. MULLEN
510
2 x 2 matrices over GF(2).
Let
i
be the cyclic group of order 2 acting on the
two rows of the 2 x 4 array
and let
2
0
1
0
1
0
0
1
1
be the cyclic group-o-order 2 acting on the 2 columns of D.
from (5.5) we have
.(2,i)
5
+ 2
Then
7 distinct classes which may also be easily
verified by direct calculation.
6., A FURTHER GENERALIZATION
In this section we consider a further generalization by allowing
i
directly on the column vectors of R rather than on the rows of the m x q
As before suppose
2
acts on the set of n columns of D.
to act
m
array.
Thus, after a matrix is
permuted by columns, it is then acted upon be a more general permutation of the
column vectors of R rather than
Just
permuting the rows of the given matrix.
example, using the example from section 5, suppose
(12...8).
8 generated by
Then if
A
i
For
is the cyclic group of order
is applied to the matrix
1
0
1
i
i
0
0
i
i
[4C7C6
we obtain the matrix
[C5C8C7]
0
1
0
0
1
1
1
1
1
which cannot be obtained from A by just permuting the rows of A.
Hence we have a
more general setting than that considered in section 5 where equivalent matrices
were obtained by simply permuting the rows and columns of the given matrix.
Suppose
2
i
is cyclic of order q
m
m
acting on the q
column vectors of R while
is cyclic of order n acting on the n columns of D.
THEOREM 6.1.
If p
2o
n
%(2’i
Im
nq
7.
t
In
@(t)q
mn/t
(6.1)
PERMUTATION MATRICES AND MATRIX EQUIVALENCE
-pln
while if
lnq
=--iml
%(2’i)
(t)qmn/t
p
Since q
b
+
x
m
PI P2
and
[
n/t
(t)
zn/t
m
t
e
p
i
(t)(pi-pi-l+l)qmn/t]
(6.2)
where p is a prime and b > i we have
q
Substituting
t.ln
t=kp
Pal(Xl,...,xq
nq
+
tin
[t#kp I
PROOF.
N
511
._Im tlq (t)xqm/tt
7.
m
m
q
7.
i=l
(pi_pi-l)
p
into (5.1) we obtain for a general term with t fixed
bm
(Zl+Z2 +’’’)
bm
(pl-p1-1)eP
+ l
i=l
bm
(z i +z
+...
2pi
P
)I
z
If t
N
qmn/(nqm) while if t’} 1 and p
(i/nqm)(t)qmn/t from which (6.1) follows
i, N
case where
pln,
if t is a divisor of n and t
as in the above case.
summing over all t
kp
If t
In yields
then N
# kp
so that
after summing over all
# kp
I
i=0
tln.
In the
for some k then N is the same
--(i/nqm)#(t)(p i -p i-I
+i) q
mn/t
so that
(6.2).
As an illustration, if q
%(2,i)
i
n then t
i
p
m
n
2 then using (6.2) we see that
3 so that the 16 2 x 2 matrices over GF(2) are decomposed into 3 dis-
joint equivalence classes.
REFERENCES
i.
Carlitz, L.
"Invariantive theory of equations in a finite field", Trans.
Amer. Math, Soc. 75 (1953), 405-427.
2.
Carlitz, L. "Invariant theory of systems of equations in a finite field",
J. Analyse Math. 3 (1953/54), 382-413.
3.
Cavior, S.R.
4.
Cavior, S.R. "Equivalence classes of sets of polynomials over a finite field",
J. fur die Reine und Angewandte Math 225 (1967), 191-202.
5.
deBruijn, N.G.
"Equivalence classes of functions over a finite field", Acta
Arith. i0 (1964), 119-136.
Pdlya’s theory of counting, Applied Combinatorial Mathematics
(ed. E.F. Beckenbach), John Wiley & Sops, New York, 1964.
512
G.L. MULLEN
Ac._t.a
6.
Mullen, G.L. "Equlvalence classes of functions over a finite fleld",
Arlth. 29 (1976), 353-358.
7.
Mullen, G.L. "Equivalence classes of polynomials over finite
Arith. 31 (1976), 113-123.
8.
Mullen, G.L. ’deak equlvalence of functions over a finite fleld’:, Acta Arlth.
35 (1978), 157-170.
9.
Mullen, G.L. "Equivalence classes of matrices over finite flelds", Lin. Alg.
& its Aps. 27 (1979), 61-68.
i0.
flelds" Acta
Mullen, G.L. "Equivalence classes of matrices over a finite fled",
Internat. J. Math. & Math. Scl. 2 (1979), 487-481.